Notes on an Endogenous Growth Model with two Capital Stocks II: The Stochastic Case

6 Summary and concluding remarks

We have proven that the functions (21) and (34) are the value functions of the social
planner and of the representative agent, respectively. We can use these functions and the
first order necessary conditions along the optimal consumption paths in order to find the
optimal level of consumption. The result is the typical consumption rule for the standard
AK model with logarithmic preferences, Cobb-Douglas technology, and full depreciation
of physical capital. It is easy to check that this result does fit the Euler equation in
consumption (31). Similarly, we can use (34) and the first order necessary condition for
the optimal human capital allocation (29). We find that the optimal way to shift human
capital between the two production sectors is to hold ut constant, once we have found
the optimal allocation. Similar to the consumption rule, it can be shown that this policy
rule fulfills the Euler equation (32). Furthermore, the restriction ut ∈ [0, 1] holds. The
transversality conditions in (33) ensure that the policy rules (35) of the representative
agent are necessary and sufficient for a utility maximizing path. In the centralized case,
the optimal stock of human capital employed in the goods sector ut is a little bit smaller
than in the decentralized case without taxation, although ut ∈ [0, 1] still holds. Hence,
the path of human capital in the centralized economy lies above the human capital path
in the decentralized economy given the same initial stocks of capital.

Finally, we have shown that the time series properties of the model are similar to
those of the standard neoclassical growth model when looking at the detrended time
series. This is due to the fact that the optimal human capital allocation is a constant
and thus unaffected by the state variables. As a consequence, the growth rate of human
capital is always equal to B(1 - ubgp). Hence, the introduction of the schooling sector
does not change the dynamics of the model.

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